The number of distinct nonlogical symbols in a language's signature, typically counting function symbols, predicate symbols and constant symbols (often treating a symbol together with its arity as a distinct entry depending on convention).

Principle

Signature size measures the vocabulary richness available to build formulas; it constrains the variety of atomic formulas and thus impacts expressiveness, the combinatorics of structures, and the complexity of decision problems.

Demonstration

A signature Σ = {c (constant), f (unary function), g (binary function), P (unary predicate), R (binary predicate)} has signature size 5 under the convention of counting symbol names; under a convention that counts symbol-arity pairs it remains 5 because arities are recorded with the symbols.

Misapplication

Counting logical connectives (¬, ∧, →) or meta-variables as signature symbols, or failing to fix whether different arities of the same name are distinct, leads to inconsistent comparisons of signature size across languages or results.

Consequence

Controlling signature size is important in finite model theory and complexity analyses: bounded signatures can simplify enumeration arguments and make some decision procedures more feasible, while unbounded or infinite signatures can increase expressiveness and technical difficulty.

Reversal

The dual view is an infinite or very large signature where new symbols are freely introduced; such expansion can trivially increase expressivity but may trivialize certain classification results or require additional bookkeeping in constructions.

Boundary

Signature size refers only to nonlogical symbol count in the chosen formal language and excludes logical connectives, punctuation, variables, and external encoding overhead; precise treatment of arity and symbol-equivalence must be stated to avoid ambiguity.

Semantic Tension

Signature size competes with vocabulary density measures like symbol arity distribution and reuse: two signatures with the same size can differ greatly if one has many high-arity symbols and the other many constants, producing different modeling capabilities.

Synthesis

Signature size is the count of distinct nonlogical symbols available in a language (with a specified convention for arity), serving as a simple metric of vocabulary richness that conditions expressiveness, combinatorics and complexity in logical investigations.